Thermoelectric device utilizing non-zero berry curvature

ABSTRACT

Thermoelectric devices and methods of using thermoelectric devices. A thermoelectric device includes a thermoelectric element comprised of a material having a non-zero Berry curvature. The device may operate as a Nernst generator that generates electricity in response to application of a temperature gradient to the thermoelectric element, or as an Ettingshausen cooler that pumps heat into or out of an object to be heated or cooled in response to application of a current to the thermoelectric element. In either application, the non-zero Berry curvature of the material allows the device to operate without an externally applied magnetic field.

This application claims the benefit of U.S. Application No. 62/570,782filed on Oct. 11, 2017 entitled “Thermoelectric Device Using WeylSemimetal”, and U.S. application Ser. No. 16/157,522 filed on Oct. 11,2018 entitled “Thermoelectric Device Utilizing Non-Zero BerryCurvature”, the disclosures of which are each incorporated by referenceherein in their entireties.

BACKGROUND

It is estimated that in 2017, the United States consumed 97.7 quadsproviding for its residential, commercial, industrial, andtransportation energy needs. Approximately 80% of this energy wasproduced by fossil fuels such as petroleum (37.1%), natural gas (28.7%),and coal (14.3%). Nuclear power (8.62%) and biomass (5.03%) accountedfor another 13.7% of the energy produced. The approximately 6% remainingwas generated from sources such as hydroelectric, wind, geothermal, andsolar. Of the energy consumed, it is further estimated that only 32%produced useful work, with the remaining 68% being rejected into theenvironment as waste heat. Transportation applications are particularlyinefficient, with about 79% of the energy consumed producing waste heat.Recovery of even a portion of this waste heat could significantly reducethe amount of energy consumed.

Attempts to recover energy from waste heat include using the heat togenerate electricity. For example, waste heat may be used to vaporizeliquids or heat gases that are provided to an engine which powers anelectric generator. Another approach to generating electricity fromwaste heat is to use the waste heat to produce a temperature gradientacross a thermoelectric device that produces electricity through athermoelectric effect.

Thermoelectric devices include thermoelectric generators, Peltierdevices, and Nernst/Ettingshausen devices. Conventional thermoelectricdevices include thermoelectric generators that generate electricity froma temperature gradient and Peltier devices such as thermoelectricheat-pumps (also referred to as Peltier coolers or thermoelectriccoolers) that use an applied current to generate a temperature gradient.Thermoelectric generators and Peltier devices have a longitudinalgeometry in which the temperature gradient and induced voltage runparallel to one another. Conventional longitudinal thermoelectricdevices require both n-type and p-type materials electrically connectedin series and thermally connected in parallel. To increase efficiency,Peltier devices are often cascaded (e.g., stacked with increasinglysmaller surface areas on top) as shown in FIG. 1. The need forelectrical connections between the n-type and p-type materials, as wellas the need for this cascading in Peltier devices, add to the cost andcomplexity of conventional, longitudinal thermoelectric devices.

Nernst/Ettingshausen devices are solid-state devices where an appliedtemperature gradient and perpendicular magnetic field generate amutually orthogonal voltage, or an applied current and perpendicularmagnetic field generate a mutually orthogonal temperature gradient. Themagnetic field generates a skew force (Lorentz force) that acceleratescharge carriers in a direction perpendicular to the temperature gradientin the device. As with Peltier devices, Nernst/Ettingshausen devicesinclude thermoelectric heat-pumps (also referred to as Ettingshausencoolers) that use an applied current to generate a temperature gradient,and thermoelectric generators (also referred to as Nernst generators)that generate electricity from a temperature gradient.Nernst/Ettingshausen devices utilize a transverse geometry and thus onlyrequire one polarity of material.

To increase efficiency, Nernst/Ettingshausen devices may be shaped asshown in FIG. 2. Because the shaping is done on the thermoelectricmaterial itself, this technology is more simplistic than that of Peltierdevices and eliminates losses due to the multiple electrical connectionsof n-type and p-type thermoelectric materials. However, unlike Peltierdevices, conventional Nernst/Ettingshausen devices require an externallyapplied magnetic field that is orthogonal to the electrical current andtemperature gradient. The magnetic fields used in conventionalNernst/Ettingshausen devices must be relatively intense, which can makeNernst/Ettingshausen devices impractical for commercial applications.

Thus, there is a need for improved thermoelectric devices and methods ofusing thermoelectric devices to provide thermoelectric electricitygeneration and/or cooling with improved efficiency that do not requiredifferent types of materials or intense externally applied magneticfields.

SUMMARY

In an embodiment of the invention, a thermoelectric device is providedcomprising a thermoelectric element including a material having anon-zero Berry curvature.

In an aspect of the invention, the thermoelectric element may beconfigured to generate a voltage in response to being exposed to atemperature gradient.

In another aspect of the invention, the thermoelectric element may beconfigured to generate a temperature gradient in response to applicationof an electrical current.

In another aspect of the invention, the non-zero Berry curvature may bealong an axis of the material orthogonal to a temperature gradient towhich the thermoelectric element is exposed or the thermoelectricelement generates.

In another aspect of the invention, the thermoelectric element may havea first side, a second side located a first distance from the first sidealong a first dimension, a third side that intersects the first andsecond sides, and a fourth side located a second distance from the thirdside along a second dimension orthogonal to the first dimension and thatintersects the first and second sides. The thermoelectric device mayfurther include a first thermal coupler configured to thermally couplethe first side to a heat source and a second thermal coupler configuredto thermally couple the second side to a heat sink. A voltage may begenerated between the third and fourth sides in response to applicationof a temperature gradient between the first thermal coupler and thesecond thermal coupler.

In another aspect of the invention, the thermoelectric device mayinclude a magnet configured to provide a magnetic field to thethermoelectric element.

In another aspect of the invention, the thermoelectric device may be oneof a Ettingshausen cooler or a Nernst generator.

In another aspect of the invention, the material may be a Weylsemimetal.

In another aspect of the invention, the Weyl semimetal may breaktime-reversal symmetry.

In another embodiment of the invention, a method of generatingelectricity is provided. The method includes providing a temperaturegradient across a thermoelectric element including a material having anon-zero Berry curvature.

In an aspect of the invention, providing the temperature gradient acrossthe thermoelectric element may include coupling the first side of thethermoelectric element to the heat source and coupling the second sideof the thermoelectric element to the heat sink, wherein the second sidemay be located the first distance from the first side along the firstdimension.

In another aspect of the invention, the method may further includeapplying the magnetic field to the thermoelectric element.

In another aspect of the invention, the method may include orienting thethermoelectric element so that the axis of the material having thenon-zero Berry curvature is orthogonal to the temperature gradient.

In another embodiment of the invention, a method of generating atemperature gradient is provided. The method includes passing a currentthrough the thermoelectric element including the material having thenon-zero Berry curvature.

In another aspect of the invention, generating the temperature gradientmay include coupling the first side of the thermoelectric element to theheat sink, and coupling the second side of the thermoelectric element toan object to be cooled or warmed, wherein the second side is located thefirst distance from the first side along the first dimension.

In another aspect of the invention, the method may include applying themagnetic field to the thermoelectric element.

In another aspect of the invention, passing the current through thethermoelectric element from the third side to the fourth side may coolthe object, and passing the current from the fourth side to the thirdside may warm the object.

In another aspect of the invention, the method may include orienting thethermoelectric element so that the axis of the material having thenon-zero Berry curvature is orthogonal to the temperature gradient andthe current.

The above summary presents a simplified overview of some embodiments ofthe invention to provide a basic understanding of certain aspects of theinvention discussed herein. The summary is not intended to provide anextensive overview of the invention, nor is it intended to identify anykey or critical elements, or delineate the scope of the invention. Thesole purpose of the summary is merely to present some concepts in asimplified form as an introduction to the detailed description presentedbelow.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate various embodiments of theinvention and, together with the general description of the inventiongiven above, and the detailed description of the embodiments givenbelow, serve to explain the embodiments of the invention.

FIG. 1 is an isometric view of a Peltier device having a cascadedconfiguration.

FIG. 2 is an isometric view of an Nernst/Ettingshausen device configuredto have a cross-sectional area that varies along the length of atemperature gradient applied across the device.

FIG. 3 is a diagrammatic view of a thermoelectric element depictingmovement of charge carriers in response to a temperature gradient acrossthe element.

FIG. 4 is an isometric view of a longitudinal thermoelectric module thatutilizes two thermoelectric elements of FIG. 3 to generate a voltagebased on the Seebeck effect.

FIG. 5A is an isometric view of a thermoelectric generator including aplurality of the thermoelectric modules of FIG. 4.

FIG. 5B is an enlarged view of a portion of the thermoelectric generatorof FIG. 5A showing additional details thereof.

FIG. 6 is an isometric view of a transverse thermoelectric element thatutilizes the Nernst effect to generate a voltage from a temperaturegradient.

FIG. 7 is an isometric view of the thermoelectric element of FIG. 6showing a spatial relationship between a crystallographic axis of thethermoelectric element along which a non-zero Berry curvature exists,the temperature gradient applied to the thermoelectric element, and thevoltage produced by the thermoelectric element.

FIG. 8 is a diagrammatic view of an exemplary transverse thermoelectricgenerator, or Nernst generator, including the thermoelectric element ofFIG. 7.

FIG. 9 is an isometric view of a thermoelectric test device used tocollect Nernst effect performance data on the thermoelectric element ofFIG. 7.

FIG. 10 is graphical view illustrating the Nernst thermopower versesapplied magnetic field at different operating temperatures.

FIG. 11 is graphical view illustrating Nernst thermopower as a functionof temperature in the absence of an applied magnetic field for athermoelectric element having a transverse geometry.

FIG. 12 is a graphical view illustrating thermoelectric figures of meritverses temperature for different thermoelectric materials in transverseand longitudinal configurations without an externally applied magneticfield.

DETAILED DESCRIPTION

Conventional thermoelectric devices use a longitudinal geometry anddepend on the Seebeck effect to generate electricity from waste heat.Thermoelectric devices having transverse geometries have significantadvantages in cost and complexity over those having a longitudinalgeometry. However, because conventional transverse thermoelectricdevices depend on the Nernst effect to generate electricity, theynormally need a large externally applied magnetic field to function.Embodiments of the invention provide the advantages of transversegeometries in thermoelectric devices having little or no need for anexternal magnetic field. This improvement in thermoelectric devices hasbeen achieved by using thermoelectric materials that have a non-zeroBerry curvature, such as found in certain Weyl semimetals.

A Weyl semimetal is a material having inverted conduction and valencebands where the bands are linear Dirac bands near the crossing points.The breaking of time-reversal symmetry or spatial-inversion symmetry maylift the degeneracy of the band crossing points, giving rise to pairs ofWeyl nodes. Separated Weyl nodes may result when the electron bandstructure of the material has singly degenerate bands that include bulkband crossings known as “Weyl points”. Electrons around the Weyl pointshave a property called Berry curvature Ω_(z) that behaves like aninternal magnetic field. The Berry curvature Ω_(z) may give electrons anadditional velocity that is normal to the direction of their momentum.One Weyl semimetal that breaks time-reversal symmetry and has a non-zeroBerry curvature is YbMnBi₂. A thermoelectric device made from YbMnBi₂ inaccordance with an embodiment of the invention has demonstratedpreviously unknown thermoelectric efficiency in a transverse geometrywithout the need for an externally applied magnetic field. Use ofmaterials having non-zero Berry curvatures thereby provides transversethermoelectric devices with significant advantages in cost andreliability over conventional devices.

Embodiments of the invention may utilize the transverse geometry andcascaded shape of conventional Nernst/Ettingshausen devices, but do notrequire an externally applied magnetic field. Weyl semimetals having anon-zero net Berry curvature may be used to form these newNernst/Ettingshausen devices. The non-zero Berry curvature may act as anintrinsic magnetic field in k-space, generating a skew force to theapplied current and thus inducing a temperature gradient. Materials witha non-zero Berry curvature may allow thermoelectric devices to be builtwith the simplicity of the Nernst/Ettingshausen geometry without theneed for an externally applied magnetic field. Thermoelectric devicesmade in accordance with embodiments of the invention also providepreviously unheard of thermoelectric figures of merit in a transversegeometry.

FIG. 3 depicts a cross-sectional view of a thermoelectric element 10comprising a length of thermoelectric material 12 having a cold end 14and a hot end 16. Application of heat to the thermoelectric element 10that produces the depicted temperature difference 4T may cause chargecarriers 18 to migrate toward and/or condense at the cold end 14 ofthermoelectric material 12 as indicated by the single headed arrows 20.This phenomenon is known as the Seebeck effect.

For a thermoelectric material 12 in which the majority carriers arepositive charge carriers 18 (e.g., a p-type semiconductor), the carriermigration may cause a positive voltage V to build up across the lengthof the thermoelectric material 12 such that the cold end 14 has a higherpotential than the hot end 16. For a thermoelectric material 12 in whichthe majority carriers are negative charge carriers 18 (e.g., an n-typesemiconductor), the carrier migration may cause a negative voltage V tobuild up across the thermoelectric material 12 such that the cold end 14has a lower potential than the hot end 16. The thermopower or Seebeckcoefficient α of the length of thermoelectric material 12 is providedby:

$\begin{matrix}{\alpha = \frac{- E}{\nabla T}} & {{Eqn}.\mspace{14mu} 1}\end{matrix}$

where E is the electric field and ∇T is the temperature gradient.Equation 1 may simplify to α=V/ΔT when the voltage V and temperaturedifference ΔT are measured over the same length of material. Thus,Equation 1 may be used to determine an expected voltage that will begenerated by the thermoelectric element 10 given the dimensions of theelement. In order to define a consistent flow of current I that isindependent of the type of charge carrier, current I is defined hereinas always moving in the direction of positive charge flow. Thus, inmaterials having negative charge carriers 18, current I flows in theopposite direction of the charge carriers 18, and in materials havingpositive charge carriers 18, the current I flows in the same directionas the charge carriers 18.

It is typically desirable to use a thermoelectric material 12 with arelatively large Seebeck coefficient α in order to generate a highervoltage V for a given temperature difference ΔT than would be generatedby a thermoelectric material with a low Seebeck coefficient α. A typicalSeebeck coefficient α for a semiconductor may have a magnitude thatranges from 200 to 300 μV/K at room temperature. The thermoelectricefficiency of a thermoelectric material can be quantified by adimensionless figure of merit zT given by:

$\begin{matrix}{{zT} = {\frac{\alpha^{2}\sigma}{\kappa}T}} & {{Eqn}.\mspace{14mu} 2}\end{matrix}$

where Tis the average absolute temperature of the thermoelectric element10 in kelvin, σ is the electrical conductivity of the thermoelectricmaterial 12, κ is the thermal conductivity of the thermoelectricmaterial 12, and each of the parameters may vary with temperature.

According to Equation 2, the thermoelectric figure of merit zT isproportional to the electrical conductivity σ and the square of theSeebeck coefficient α, and inversely proportional to the thermalconductivity κ. A low thermal conductivity κ may enable a temperaturegradient ∇T to be maintained across the thermoelectric element 10 with alower flow of heat through the thermoelectric element 10 as compared toa high thermal conductivity κ. A high electrical conductivity σ maylower the impedance of the thermoelectric element 10, thereby allowingit to source a larger amount of current I as compared to athermoelectric element 10 with a low electrical conductivity σ.

It is normally desirable to use materials with as high of athermoelectric figure of merit zT as possible. Useful devices may bemade from thermoelectric materials with a thermoelectric figure of meritof 0.3. In contrast, a thermoelectric figure of merit of 1.0 isconsidered good, and a thermoelectric figure of merit of 2.0 or more isconsidered to be near the limit of what is possible with conventionaltechnology. As can be seen from Equations 1 and 2, the voltage Vgenerated by thermoelectric element 10 from a given temperaturedifference ΔT due to the Seebeck effect is limited by the intrinsicproperties of the thermoelectric material 12, e.g., the Seebeckcoefficient α, electrical conductivity σ, and thermal conductivity κ.

The temperature difference ΔT may be generated by the thermoelectricelement 10 itself in response to a current I being driven through thethermoelectric element 10. Thus, the thermoelectric element 10 can beused as a heat pump by passing an externally sourced current I throughthe element. To generate the shown temperature difference ΔT in athermoelectric material in which the majority carriers are positivecharge carriers (e.g., holes), the current I may be driven from the coldend 14 toward the hot end 16. In contrast, to generate the showntemperature difference ΔT in a thermoelectric material in which themajority carriers are negative charge carriers (e.g., electrons), thecurrent I may be driven from the hot end 16 toward the cold end 14.

FIG. 4 depicts a thermoelectric module 22 which utilizes the Seebeckeffect and has a longitudinal geometry. Thermoelectric module 22 mayinclude a thermoelectric element 24 made of a p-type thermoelectricmaterial having a hot end 26 and a cold end 28, a thermoelectric element30 made of an n-type thermoelectric material having a hot end 32 and acold end 34, an upper electrode 36 coupling the cold end 28 ofthermoelectric element 24 to the cold end 34 of thermoelectric element30, and lower electrodes 38 coupling the hot ends 26, 32 ofthermoelectric elements 24, 30 to an electrical load 42. Because thedepicted type of thermoelectric module 22 relies on the Seebeck effectto generate electrical power, it uses thermoelectric elements 24, 30having different types of charge carriers and has an electrical outputthat scales intrinsically with the properties of the thermoelectricmaterials used in the thermoelectric elements 24, 30.

Application of a temperature difference ΔT across the thermoelectricmodule 22 may cause heat to flow from the hot ends 26, 32 to the coldends 28, 34 of thermoelectric elements 24, 30, as indicated by singleheaded arrows 44, 46. The resulting temperature gradient ∇T may cause aflow of positive charge carriers from the hot end 26 to the cold end 28of thermoelectric element 24 that results in a positive current flowtoward the cold end 28, as indicated by single headed arrow 48. Thetemperature gradient ∇T may also cause a flow of negative chargecarriers from the hot end 32 to the cold end 34 of thermoelectricelement 30 that results in a positive current flow toward the hot end 32as indicated by single headed arrow 50. The electrodes 36, 38 may beconfigured to complete the circuit, thereby allowing current to flowthrough the thermoelectric module 22 and electrical load 42.

Because the voltage generated by thermoelectric module 22 is limited bythe intrinsic properties of the thermoelectric materials from which itis made, thermoelectric generators using a longitudinal geometry aretypically assembled from a large number of modules in order to produce auseful output voltage. FIGS. 5A and 5B depict an exemplarythermoelectric generator 52 including a plurality of thermoelectricmodules 22 electrically coupled in a series configuration. The upperelectrodes 36 of thermoelectric modules 22 are thermally coupled to anupper substrate 54, and the lower electrodes 38 of thermoelectricmodules 22 are thermally coupled to a lower substrate 56. Thethermoelectric modules 22 are thus thermally coupled to the upper andlower substrates 54, 56 in parallel. The substrates 54, 56 may be madefrom a material that has a low electrical conductivity, or iselectrically isolated from the electrodes 36, 38 by an electricallyinsulating layer (not shown), in order to avoid shorting out thethermoelectric generator 52.

One of the substrates 54, 56 (e.g., the lower substrate 56) may bethermally coupled to a heat source 58 (e.g., the exhaust from combustionused to heat a boiler), and the other substrate 54, 56 (e.g., the uppersubstrate 54) may be thermally coupled to a heat sink 60 (e.g., acooling medium such as the atmosphere or a reservoir of water).Thermally coupling the thermoelectric generator 52 between a heat sourceand a heat sink as described above may cause a temperature difference ∇Tdevelop across each of the thermoelectric modules 22. The resultingtemperature gradient ∇T in thermoelectric elements 24, 30 may in turncause each thermoelectric module 22 to generate a voltage. Theelectrodes 36, 38 of thermoelectric modules 22 may be configured toelectrically couple the thermoelectric modules 22 in a seriesconfiguration so that the voltages generated by the thermoelectricmodules 22 add constructively to generate an output voltage V thatcauses a current I to flow through an electrical load 62 coupled tooutput terminals 64, 66 of thermoelectric generator 52.

FIG. 6 depicts a thermoelectric element 70 having a transverse geometrythat utilizes the Nernst effect to generate a voltage from a temperaturedifference ΔT. The thermoelectric element 70 may include a heightdimension h that generally corresponds with an x-axis of athree-dimensional coordinate system 72, a length dimension l thatgenerally corresponds with a y-axis of coordinate system 72, and a widthdimension w that generally corresponds with a z-axis of coordinatesystem 72.

To generate a voltage V across the length l of thermoelectric element70, the Nernst effect requires a force that urges charge carriers in adirection orthogonal (i.e., perpendicular) to the temperature gradient∇T produced by temperature difference ΔT across the element, e.g.,orthogonal to the height h of thermoelectric element 70. In conventionaldevices, this force is a Lorentz force resulting from the cross-productof the temperature gradient ∇T and a magnetic field H applied in adirection orthogonal to both the temperature gradient ∇T and the voltagegradient ∇V generated by the device. For the thermoelectric element 70depicted in FIG. 6, this magnetic field H may be generally parallel tothe z-axis of coordinate system 72.

As carriers move toward the cold side of the thermoelectric element 70under the influence of the temperature gradient ∇T (as indicated bysingle headed arrow 71), the magnetic field H may generate forces on thecarriers that urge positive and negative charge carriers in oppositedirections. For the depicted temperature difference ΔT and magneticfield H, this force may urge positive charge carriers in a positivedirection along the y-axis and negative charge carriers in a negativedirection along the y-axis. The movement of the charge carriers inthermoelectric element 70 under the influence of the temperaturegradient ∇T and magnetic field H may thereby produce a voltage V acrossthe thermoelectric element 70 having the shown polarity.

In contrast to the output of thermoelectric module 22, which is limitedby the intrinsic properties of the materials from which it is made, theoutput of thermoelectric element 70 scales with the size of the device.Advantageously, this allows the voltage V generated by thermoelectricelement 70 to be scaled by simply adjusting its dimensions. Thus,thermoelectric generators based on this type of element may avoid thecomplexity of assembling a large number of thermoelectric elements asshown in FIG. 5. However, because the magnetic field H required togenerate useful amounts of electricity is typically quite large,conventional thermoelectric elements utilizing a transverse geometry andthe Nernst effect are generally not suitable for power recovery fromwaste heat or other practical commercial applications.

Embodiments of the invention advantageously reduce or eliminate the needto provide a magnetic field H to thermoelectric elements utilizing theNernst effect by using materials having a non-zero Berry curvature, suchas certain Weyl semimetals. A Weyl semimetal is a solid-state crystalhaving low energy excitations that comprise Weyl fermions which carryelectrical charge. It has been determined that Weyl semimetals having anon-zero integral over the Fermi surface of the projection of the Berrycurvature Ω_(z) of the dispersion relation of their conduction electronsalong a specific crystallographic axis can be used to create or increaseNernst thermopower α_(xyz).

This property may enable materials having a non-zero Berry curvature togenerate a thermoelectric voltage along a direction orthogonal to thedirection of an applied temperature gradient without an externallyapplied magnetic field. Thermoelectric devices made using materialshaving a non-zero Berry curvature may be used to generate power and/orpump heat, and thus may have wide-ranging applications in manyindustries, such as the energy and electronics industries.

The origin of this thermoelectric effect is believed to lie in thepresence of a non-zero Berry curvature of the electronic band structureat each electron energy and momentum value. A Berry curvature may beproduced by an electronic band structure that exhibits spin-orbitcanting, which may cause the material having the canted spin-orbit toexhibit a non-zero magnetic moment. The ability of a Weyl semimetal togenerate voltages as described above may depend on the integral of theprojection of the Berry curvature over the Fermi surface being non-zero.One way this may occur is when electrons in the solid breaktime-reversal symmetry. The Weyl nodes act as monopole sources (orsinks) of the Berry curvature. This Berry curvature acts as an effectivemagnetic field that exists in the electrons' momentum-space, introducingan anomalous velocity to electron motion that is skew to both the Berrycurvature and the electrons' momentum. This skew force is believed togenerate a non-zero thermoelectric power in a direction orthogonal toboth the net Berry curvature integrated over the whole Fermi surface andthe direction of the applied temperature gradient. This effect has beenobserved experimentally in the compound YbMnBi₂, which is a Weylsemimetal that breaks time-reversal symmetry and is a cantedantiferromagnet material having a net Berry curvature Ω_(z) along its[110] crystal axis.

By aligning the [110] crystal axis of a YbMnBi₂ crystal with the z-axisas depicted in FIG. 7, the above described effect may be used to producetransverse thermoelectric devices, namely Nernst generators and theirthermodynamic reciprocal, Ettingshausen coolers. Unlike classical Nernstgenerators and Ettingshausen coolers, thermoelectric devices made withmaterials having a non-zero Berry curvature do not require an externalmagnetic field H, the role of which is provided by the Berry curvatureΩ_(z). An external magnetic field can, however, still be applied, andits presence can be used to adjust (e.g., reinforce or counteract) theeffect of the Berry curvature in certain circumstances.

Transverse thermoelectric devices such as shown in FIG. 7 have severaladvantages over classical Peltier devices, such as the thermoelectricmodules and generator shown in FIGS. 4 and 5. Because the electrodesapplied to the thermoelectric material can lie in an isothermal plane ofa transverse device, the electrodes can both be applied at either thehot end or the cold end of the device. That is, there is no need toreturn the current from the hot end to the cold end of the device. Thus,there is no need for a thermocouple pair, with a p-type materialcarrying the current from hot to cold and an n-type material carryingthe current back from cold to hot.

Transverse thermoelectric devices have several advantages overconventional, longitudinal thermoelectric devices. For example, there isno need to simultaneously develop n-type and p-type thermoelectricmaterials with similar temperature dependences in their zT values. Onematerial with one polarity suffices for the entire device. Anotheradvantage is that there is no need to connect several thermocouplestogether electrically in series and thermally in parallel as must bedone in thermoelectric modules and Peltier devices. Rather, in Nernstgenerators and Ettingshausen coolers, the current capacity and thevoltage rating can be increased by simply increasing the physical sizeof the thermoelectric material in the device. More advantageously,having one pair of current contacts and one pair of thermal contacts mayallow the parasitic losses in contact resistances of thermoelectricmodules and Peltier devices to be decreased as compared to Seebeckeffect-based devices using series electrical couplings.

The temperature difference in classical Peltier devices may be limitedby the following equation:

ΔT _(max)=½×(zT×T _(cold))  Eqn. 3

Therefore, when larger temperature drops are required in a Peltierdevice, several Peltier elements are typically connected in cascadedcoolers. This increases the complexity of devices aimed at generatingcooling over large temperature gradients, especially in cryogeniccooling applications. However, these limitations in the maximumtemperature gradient do not hold for Nernst/Ettingshausen devices.Therefore, Nernst/Ettingshausen devices can operate with largetemperature gradients without cascading series connections.

The large transverse Nernst thermopower α_(xyz) of Weyl semimetals hasbeen demonstrated experimentally using YbMnBi₂. The peak in the Nernstthermopower α_(xyz) of YbMnBi₂ in the absence of a magnetic field isapproximately 1000 μV/K near 50K and 30 μV/K near room temperature. Byway of comparison, commercially available thermoelectric materials havea Seebeck coefficient α_(xxz) near 200-300 μV/K at room temperature. Thetransverse thermoelectric figure of merit zT may be calculated usingEquation 4 below:

$\begin{matrix}{{zT_{xy}} = {\frac{\left( \alpha_{xyz} \right)^{2} \times \sigma_{yy}}{\kappa_{xx}} \times T}} & {{Eqn}.\mspace{14mu} 4}\end{matrix}$

where α_(xyz) is the Nernst thermopower, σ_(yy) is the electricalconductivity in the direction parallel to the measured voltage, andκ_(xx) is the thermal conductivity in the direction parallel to theapplied and measured temperature gradient.

For YbMnBi₂, the transverse zT in the absence of a magnetic field isestimated to be 2.42 at 59.55 K. This value is greater than that of anyother known thermoelectric for this low of a temperature range, and fora transverse geometry (i.e. Nernst or Ettingshausen geometry) at anytemperature. The performance of this new thermoelectric material, whichis based on new physical principles, enables the novel approach tosolid-state cryogenic cooling provided by embodiments of the invention,and may be useful in numerous cooling applications. For example, anumber of detectors including infra-red detectors, focal plane arrays,and x-ray and gamma-ray detectors, could benefit from Ettingshausencoolers using thermoelectric elements having a non-zero Berry curvature.

FIG. 8 depicts an exemplary thermoelectric device 80 in accordance withan embodiment of the invention. The thermoelectric device 80 may includea thermoelectric element 82 comprising a material having a non-zeroBerry curvature, a thermal coupler 84 configured to couple a hot side 86of the thermoelectric element 82 to a heat source 88, and a thermalcoupler 90 configured to couple a cold side 92 of the thermal element toa heat sink 94. The thermoelectric element 82 may be configured so thatan axis of the material having the non-zero Berry curvature is generallyorthogonal to the temperature and voltage gradients. Voltage outputsides 96, 98 of thermoelectric element 82 may be electrically coupled torespective terminals 100, 102 to facilitate connection of thethermoelectric device 80 to an electrical load 104. The thermoelectricdevice 80 may also include one or more magnets 106, 108 configured toprovide a magnetic flux 110 to the thermoelectric element 82. Themagnetic flux 110 may enter and exit the thermoelectric element 82through the remaining sides 112, 114, and may be generally aligned withthe axis having the non-zero Berry curvature. The magnetic flux 110 maybe used to adjust (e.g., reinforce or counteract) the effect of theBerry curvature, thereby providing a mechanism for controlling theoutput and/or efficiency of the thermoelectric device 80.

EXPERIMENTAL RESULTS

FIG. 9 depicts a test device 120 including a thermoelectric element 122comprising YbMnBi₂. The thermoelectric element 122 is affixed to asilicon substrate 124. The thermoelectric element 122 includes anoutward facing surface 126 having a rectangular shape that faces awayfrom the substrate 124, and a downward facing surface (not shown)generally parallel to and spaced about 0.44 mm from the outward facingsurface 126. The thermoelectric element 122 is oriented so that the[110] crystal axis of the YbMnBi₂ (and thus the non-zero Berrycurvature) is orthogonal (i.e., normal) to the outward and downwardfacing surfaces, e.g., projecting outward from the outward facingsurface 126. The rectangular shape of outward facing surface 126 isdefined by a left facing surface 127, a right facing surface 128generally parallel to and about 2.58 mm from the left facing surface127, a top facing surface 129 that intersects the left and right facingsurfaces 127, 128, and a bottom facing surface 130 generally parallel toand 1.85 mm from the top facing surface 129. Each of the surfaces127-130 is generally orthogonal to the outward facing and downwardfacing surfaces so that the thermoelectric element 122 generally forms apolyhedron having six sides and dimensions of 0.44 by 1.85 by 2.58 mm.

The left facing surface 127 of thermoelectric element 122 is thermallycoupled to a resistive heater 136 by a copper foil heat spreader 138.The heat spreader 138 is configured to provide heat generated by theresistive heater 136 evenly to the left facing surface 127 ofthermoelectric element 122. The right facing surface 128 ofthermoelectric element 122 is coupled to a copper foil heat sink 140.When the resistive heater 136 is energized, a temperature gradient formshaving a decreasing temperature across the thermoelectric element 122from the left facing surface 127 to the right facing surface 128. Agold-plated copper bracket 142 attached to (e.g., epoxied to) the heatsink 140 and substrate 124 holds the test device 120 in place withrespect to the substrate 124. A clamp 146 holds the bracket 142 in placerelative to a base 144 so that the substrate 124 is suspended above thebase 144. The bracket 142 is in thermal contact with the heat sink 140,and thermally and electrically isolated from the base 144. Insulatedcopper leads 148, 150 electrically couple the top and bottom surfaces129, 130 to respective terminals 152, 154 that facilitate measuringvoltages V and/or currents I generated by or provided to thethermoelectric element 122.

FIG. 10 depicts a graph 160 including plots 162-168 of the Nernstthermopower α_(xyz) in μV/K verses magnetic field strength H in Oersteds(Oe). The data used to define plots 162-168 was measured using the testdevice 120 at sample temperatures of 15.86 K (plot 162), 23.17 K (plot163), 41.48 K (plot 164), 59.44 K (plot 165), 75.15 K (plot 166), 118.4K (plot 167), and 323.1 K (plot 168). The plots 162-168 illustrate theNernst Effect in YbMnBi₂. The magnetic field was applied along an axisorthogonal to the outward facing surface 126 of thermoelectric element122, which is aligned with the [110] crystallographic axis and is thusthe direction of the expected non-zero Berry curvature. Positive valuesof H indicate the field is oriented in an outward facing direction withrespect to the outward facing surface 126 as depicted in FIG. 8. As canbe seen from plot 128, the Nernst thermopower α_(xyz) has a non-zerovalue in the absence of an externally applied magnetic field that issignificantly larger than that produced by conventional thermoelectricmaterials.

FIG. 11 depicts a scatter plot 180 including data points 182illustrating the Nernst thermopower α_(xyz) in μV/K verses thetemperature of the thermoelectric element 122 in kelvin for a transversegeometry (i.e., produced by the Nernst effect), and data points 184illustrating the Seebeck coefficient α_(xxz) μV/K verses the temperatureof the thermoelectric element 122 in kelvin for a longitudinal geometry(i.e., produced by the Seebeck effect) in the absence of an externallyapplied magnetic field. The data points 182, 184 were extracted fromdata taken without an externally applied magnetic field. In the absenceof a Berry curvature, the Nernst thermopower α_(xyz) indicated by datapoints 182 would be expected to be 0 μV/K at all temperatures. Thus, thenon-zero values shown for the Nernst thermopower α_(xyz) indicate thepresence of a Berry curvature and confirm the effect of the Berrycurvature on the Nernst thermopower α_(xyz), which is significantlylarger than would be expected for conventional thermoelectric materialsconfigured in a transverse geometry. The disparity between thecoefficients α_(xyz) and α_(xxy) produced by the Nernst and Seebeckeffects support the conclusion that a non-zero Berry curvature producesthe Nernst effect even in the absence of an externally applied magneticfield.

The conventional thermoelectric figure of merit, zT, may be calculatedusing Equation 5 below:

$\begin{matrix}{{zT} = {{\frac{\alpha_{xx}^{2} \times \sigma_{xx}}{\kappa_{xx}}T} = \frac{\alpha_{xx}^{2}}{\kappa_{xx}\rho_{xx}}}} & {{Eqn}.\mspace{14mu} 5}\end{matrix}$

where ρ_(xx) is the resistivity of the thermoelectric material. Themaximum value of zT known by the Applicant to have been measured in alaboratory is 2.2 at 915 K. The transverse figure of merit zT may becalculated using Equation 6:

$\begin{matrix}{{zT_{xy}} = {{\frac{\alpha_{xy}^{2} \times \sigma_{yy}}{\kappa_{xx}}T} = {\frac{\alpha_{xy}^{2}}{\kappa_{xx}\rho_{yy}} = \frac{\alpha_{xy}^{2}}{\kappa_{xx}\rho_{xx}}}}} & {{Eqn}.\mspace{14mu} 6}\end{matrix}$

FIG. 12 depicts a scatter plot 190 including data points 192-194 showingfigures of merit zT for different thermoelectric materials in theabsence of an external magnetic field. Data points 192 illustrate thetransverse figure of merit zT verses temperature in kelvin for YbMnBi₂in a Nernst/Ettingshausen configuration. Data points 193 illustrate thelongitudinal figure of merit zT verses temperature in kelvin for PbTedoped with 2 mol % Na and nanostructured with 4 mol % SrTe. Data points194 illustrate the longitudinal figure of merit zT verses temperature inkelvin for commercially available Bi₂Te₃.

The depicted properties of YbMnBi₂ provide experimental support for theimproved performance of thermoelectric devices in accordance withembodiments of the invention. Use of materials having a non-zero Berrycurvature, such as YbMnBi₂, in fabricating thermoelectric devices allowsa simpler transverse geometry that operates without an externallyapplied magnetic field. Moreover, the peak zT of YbMnBi₂ occurs in atemperature range appropriate for cooling (e.g., zT=2.42 at 59.55K).Thus, thermoelectric devices using materials such as YbMnBi₂ may besuitable for use in both power recovery applications and for heat pumpsthat cryogenically cool electronic devices.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the embodimentsof the invention. As used herein, the singular forms “a”, “an” and “the”are intended to include the plural forms as well, unless the contextclearly indicates otherwise. It will be further understood that theterms “comprises” and/or “comprising,” when used in this specification,specify the presence of stated features, integers, actions, steps,operations, elements, and/or components, but do not preclude thepresence or addition of one or more other features, integers, actions,steps, operations, elements, components, and/or groups thereof.Furthermore, to the extent that the terms “includes”, “having”, “has”,“with”, “comprised of”, or variants thereof are used in either thedetailed description or the claims, such terms are intended to beinclusive in a manner similar to the term “comprising”.

While all the invention has been illustrated by a description of variousembodiments, and while these embodiments have been described inconsiderable detail, it is not the intention of the Applicant torestrict or in any way limit the scope of the appended claims to suchdetail. Additional advantages and modifications will readily appear tothose skilled in the art. The invention in its broader aspects istherefore not limited to the specific details, representative apparatusand method, and illustrative examples shown and described. Accordingly,departures may be made from such details without departing from thespirit or scope of the Applicant's general inventive concept.

1-20. (canceled)
 21. A method of generating electricity comprising:providing a temperature gradient across a first dimension of athermoelectric element including a material having a non-zero Berrycurvature along the first dimension; generating a voltage gradient alonga second dimension of the thermoelectric element aligned with across-product of the temperature gradient and the non-zero Berrycurvature; and coupling a voltage provided by the voltage gradient to anelectrical load.
 22. The method of claim 21 wherein providing thetemperature gradient across the first dimension of the thermoelectricelement comprises: coupling a first side of the thermoelectric elementto a heat source; and coupling a second side of the thermoelectricelement to a heat sink, the second side located a first distance fromthe first side along a third dimension transverse to both the firstdimension and second dimension.
 23. The method of claim 22 furthercomprising: applying a magnetic field aligned with the non-zero Berrycurvature to the thermoelectric element.
 24. The method of claim 21wherein the material is a Weyl semimetal.
 25. The method of claim 21wherein the temperature gradient is orthogonal to the non-zero Berrycurvature.
 26. A method of pumping heat into or out of a thermal load,comprising: passing a current through a thermoelectric element includinga material having a non-zero Berry curvature along a first dimensionsuch that the current flows across the first dimension; generating atemperature gradient along a second dimension of the thermoelectricelement aligned with a cross-product of the current and the non-zeroBerry curvature; and coupling the temperature gradient to the thermalload.
 27. The method of claim 26 wherein coupling the temperaturegradient to the thermal load comprises: coupling a first side of thethermoelectric element to a heat sink; and coupling a second side of thethermoelectric element to an object to be cooled or warmed, the secondside located a first distance from the first side along the seconddimension.
 28. The method of claim 27 wherein passing the currentthrough the thermoelectric element in a first direction cools theobject, and passing the current in a second direction opposite the firstdirection warms the object.
 29. The method of claim 26 furthercomprising: applying a magnetic field aligned with the non-zero Berrycurvature to the thermoelectric element.
 30. The method of claim 26wherein the material is a Weyl semimetal.
 31. The method of claim 26wherein the non-zero Berry curvature is orthogonal to both thetemperature gradient and the current.
 32. A thermoelectric devicecomprising: a thermoelectric element including a material having anon-zero Berry curvature along a first dimension of the thermoelectricelement; a first thermal coupler thermally coupled to a first side ofthe thermoelectric element; a second thermal coupler thermally coupledto a second side of the thermoelectric element, the second side locateda first distance from the first side along a second dimension transverseto the first dimension; a first terminal electrically coupled to a thirdside of the thermoelectric element; and a second terminal electricallycoupled a fourth side of the thermoelectric element, the fourth sidelocated a second distance from the third side along a third dimensionaligned with a cross-product of the second dimension and the non-zeroBerry curvature.
 33. The thermoelectric device of claim 32 wherein thethermoelectric element generates a voltage across the first and secondterminals in response to an application of a temperature gradient acrossthe first and second thermal couplers.
 34. The thermoelectric device ofclaim 32 wherein the thermoelectric element generates a temperaturegradient across the first and second thermal couplers in response to anapplication of an electrical current through the first and secondterminals.
 35. The thermoelectric device of claim 32 wherein thenon-zero Berry curvature is orthogonal to a temperature gradient towhich the thermoelectric element is exposed or the thermoelectricelement generates.
 36. The thermoelectric device of claim 32 wherein:the first thermal coupler is configured to thermally couple the firstside to a heat source; and the second thermal coupler is configured tothermally couple the second side to a heat sink, wherein a voltage isgenerated between the third and fourth sides in response to applicationof a temperature gradient between the first thermal coupler and thesecond thermal coupler.
 37. The thermoelectric device of claim 32further comprising: a magnet configured to provide a magnetic fieldaligned with the non-zero Berry curvature to the thermoelectric element.38. The thermoelectric device of claim 32 wherein the material is a Weylsemimetal.
 39. The thermoelectric device of claim 38 wherein the Weylsemimetal breaks time-reversal symmetry.